A System of Simultaneous Pell Equations
Added: from Cipu and Mignotte, the first system expected to have two positive solutions, one piece $\color{red}{x^2 - 24 z^2 = 1},$ uses $490^2 - 1 = 240099.$ $$ 5^2 - 24 \cdot 1^2 = 1 \; , \; \; \; 490^2 - 240099 \cdot 1^2 = 1 $$ $$ 4801^2 - 24 \cdot 980^2 = 1 \; , \; \; \; 480199^2 - 240099 \cdot 980^2 = 1 $$
The first system expected to have two positive solutions, one piece $\color{red}{x^2 - 48 z^2 = 1},$ uses $1358^2 - 1 = 1844163.$ $$ 7^2 - 48 \cdot 1^2 = 1 \; , \; \; \; 1358^2 - 1844163 \cdot 1^2 = 1 $$ $$ 18817^2 - 48 \cdot 2716^2 = 1 \; , \; \; \; 3688327^2 -1844163 \cdot 2716^2 = 1 $$ $$ $$ $$ $$
I ran the thing for my $j \leq 1148961$ with $k \leq 1000000,$ or $$ x \approx y \approx \color{magenta}{10^{1143895}} $$ no matches
Probably the most accurate way to search, just integer arithmetic, with a little cleverness to keep track of comparable subscripts $j,k$ so as not to waste time comparing $x_j$ and $y_k$ when the ratio cannot be close to $1.$
Second try: as there are just two sequences, in order to avoid saving long stretches, we could just run both sequences up a few terms. Then, if the current $x_j < y_k,$ take more terms $x_j$ until we have exceeded the stationary $y_k.$ Next, increase the $k$ until $y_k$ is bigger. The computer will not need any array, just three x terms and three y terms.
You are asking whether these two sequences have any elements larger than $1$ in common:
$$ 0, 1, 10, 99, 980, 9701, 96030, ... $$ $$ x_{j+2} = 10 x_{j+1} - x_j $$
$$ 0, 1, 14, 195, 2716, 37829, 526890, ... $$ $$ y_{k+2} = 14 y_{k+1} - y_k $$
================================
Thu May 3 10:02:52 PDT 2018
j 3 x 99
k 3 y 195
j 4 x 980
k 4 y 2716
j 5 x 9701
k 5 y 37829
j 6 x 96030
k 6 y 526890
j 7 x 950599
k 7 y 7338631
j 8 x 9409960
k 8 y 102213944
j 10 x 922080050
k 9 y 1423656585
j 11 x 9127651499
k 10 y 19828978246
j 12 x 90354434940
k 11 y 276182038859
j 13 x 894416697901
k 12 y 3846719565780
j 14 x 8853812544070
k 13 y 53577891882061
j 15 x 87643708742799
k 14 y 746243766783074
j 16 x 867583274883920
k 15 y 10393834843080975
j 18 x 85014307126080090
k 16 y 144767444036350576
j 19 x 841554882220704499
k 17 y 2016350381665827089
j 20 x 8330534515080964900
k 18 y 28084137899285228670
j 21 x 82463790268588944501
k 19 y 391161580208327374291
j 22 x 816307368170808480110
k 20 y 5448177985017298011404
j 23 x 8080609891439495856599
k 21 y 75883330210033844785365
j 24 x 79989791546224150085880
k 22 y 1056918444955456528983706
j 26 x 7838183264161795899936130
k 23 y 14720974899166357560986519
j 27 x 77590015336047156994359099
k 24 y 205036730143373549324827560
j 28 x 768061970096309774043654860
k 25 y 2855793247108063332986599321
j 29 x 7603029685627050583442189501
k 26 y 39776068729369513112487562934
j 30 x 75262234886174196060378240150
k 27 y 554009168964065120241839281755
j 31 x 745019319176114910020340211999
k 28 y 7716352296767542170273262381636
j 33 x 73004290249573634131409898586401
k 29 y 107474922985781525263583834061149
j 34 x 722667971538861366409955961984170
k 30 y 1496932569504173811519900414474450
j 35 x 7153675425139040029968149721255299
k 31 y 20849581050072651836015021968581151
j 36 x 70814086279851538933271541250568820
k 32 y 290397202131512951892690407145661664
j 37 x 700987187373376349302747262784432901
k 33 y 4044711248791108674661650678070682145
j 38 x 6939057787453911954094201086593760190
k 34 y 56335560280944008493370419085843888366
j 39 x 68689590687165743191639263603153168999
k 35 y 784653132684425010232524216523743754979
j 41 x 6730878900154869456431345085846226129001
k 36 y 10928808297301006134761968612246568681340
j 42 x 66628832152464491044351152423517323360210
k 37 y 152218663029529660876435036354928217783781
j 43 x 659557442624490040987080179149327007473099
k 38 y 2120132474116114246135328540356748480291594
j 44 x 6528945594092435918826450639069752751370780
k 39 y 29529635974596069785018164528639550506298535
j 45 x 64629898498299869147277426211548200506234701
k 40 y 411294771170228862744118974860596958607887896
j 46 x 639770039388906255553947811476412252310976230
k 41 y 5728597160408608008632647483519717870004132009
j 47 x 6333070495390762686392200688552574322603527599
k 42 y 79789065474550283258112945794415453221449960230
j 49 x 620576278649796443397288390051940735414639470001
k 43 y 1111318319483295357604948593638296627230295311211
j 50 x 6143071851583445713364515841445358023172670400250
k 44 y 15478667407291584723211167365141737328002684396724
j 51 x 60810142237184660690247870024401639496312064532499
k 45 y 215590025382598890767351394518346025964807286242925
j 52 x 601958350520263161189114184402571036939947974924740
k 46 y 3002781687949092886019708355891702626179299323004226
j 53 x 5958773362965446951200893974001308729903167684714901
k 47 y 41823353605904701513508565587965490740545383235816239
j 54 x 58985775279134206350819825555610516262091728872224270
k 48 y 582524168794716728303100209875625167741456065978423120
j 55 x 583898979428376616556997361582103853891014121037527799
k 49 y 8113515009520129494729894372670786857639839540462107441
j 57 x 57216141210617942975634540541072176372589480693993009401
k 50 y 113006685964487096197915421007515390839216297500491081054
j 58 x 566381408087174797797126251620456335703246757458427040290
k 51 y 1573980088493299217276085999732544684891388325466413027315
j 59 x 5606597939661130034995627975663491180659878093890277393499
k 52 y 21922714552941701945667288575248110197640220259029291301356
j 60 x 55499597988524125552159153505014455470895534181444346894700
k 53 y 305344023652690528022065954053740998082071695300943665191669
j 61 x 549389381945580125486595907074481063528295463720553191553501
k 54 y 4252893616584725690363256068177125862951363513954182021382010
j 62 x 5438394221467277129313799917239796179812059103024087568640310
k 55 y 59235166608533469137063519000426021083237017500057604634156471
j 64 x 532907134105804634547200232735995011166110896562179137379855680
k 56 y 825039438902883842228526009937787169302366881486852282856808584
j 65 x 5275236788225319154304350924094626630926516670055271051303707201
k 57 y 11491316978031840322062300620128594349149899323315874355361163705
j 66 x 52219460748147386908496309008210271298099055803990531375657216330
k 58 y 160053398253542880666643682671862533718796223644935388692199483286
j 67 x 516919370693248549930658739158008086350064041369850042705268456099
k 59 y 2229256258571568489010949256785946877713997231705779567335431602299
j 68 x 5116974246184338112398091082571870592202541357894509895677027344660
k 60 y 31049534221748415965486645912331393754277165020235978554003842948900
j 69 x 50652823091150132574050252086560697835675349537575248914065004990501
k 61 y 432464222845906255027802093515853565682166313051597920188718369682301
j 70 x 501411256665316987628104429783035107764550954017857979244973022560350
k 62 y 6023449585620939154423742663309618525796051217702134904088053332603314
j 72 x 49133186178954880449441836027654868690333790952392187456111679183569640
k 63 y 83895829975847241906904595192818805795462550734778290737044028286764095
j 73 x 486368402045986784750711366230804896523528075333280870017581126615083401
k 64 y 1168518170076240447542240590036153662610679659069193935414528342682094016
j 74 x 4814550834280912967057671826280394096544946962380416512719699586967264370
k 65 y 16275358551091519023684463665313332470754052676233936805066352769262552129
j 75 x 47659139940763142885826006896573136068925941548470884257179414743057560299
k 66 y 226686501545205025884040250724350500927946057808205921335514410426993635790
j 76 x 471776848573350515891202397139450966592714468522328426059074447843608338620
k 67 y 3157335663081778843352879046475593680520490756638648961892135393208648348931
j 77 x 4670109345792742016026197964497936529858218743674813376333565063693025825901
k 68 y 43976012781599698781056266399933961026358924535132879545154381094494083249244
j 78 x 46229316609354069644370777247839914331989472968225805337276576189086649920390
k 69 y 612506843279314004091434850552599860688504452735221664670269199929708517140485
j 80 x 4530001250868125474632444967891172153568375636417606594627045392082648083859600
k 70 y 8531119793128796358499031641336464088612703413757970425838614417921425156717546
j 81 x 44842389451933506791896768104397820328893719853237482706274021723999307365218001
k 71 y 118823170260523835014895008128157897379889343339876364297070332650970243676905159
j 82 x 443893893268466942444335236076087031135368822895957220468113171847910425568320410
k 72 y 1654993263854204893850031082152874099229838103344511129733146042695661986319954680
j 83 x 4394096543232735917651455592656472491024794509106334721974857696755104948317986099
k 73 y 23051082523698344678885540142012079491837844103483279451966974265088297564802460361
j 84 x 43497071539058892234070220690488637879112576268167389999280463795703139057611540580
k 74 y 321060162067922620610547530906016238786499979345421401197804493668540503920914490374
j 85 x 430576618847356186423050751312229906300100968172567565270829780260276285627797419701
k 75 y 4471791186427218343868779892542215263519161866732416337317295937094478757328000404875
j 87 x 42192114550497673533541322173005874344918870086402515061819343607810320886575829144599
k 76 y 62284016447913134193552370964684997450481766154908407321244338625654162098671091177876
j 88 x 417658876388042232363416784437626933024066803758567642355484418739296149148537928789560
k 77 y 867504439084356660365864413613047749043225564301985286160103444822063790624067276085389
j 89 x 4134396649329924650100626522203263455895749167499273908493024843785151170598803458751001
k 78 y 12082778130733080110928549419617983489154676134072885598920203888883238906638270774017570
j 90 x 40926307616911204268642848437595007625933424871234171442574764019112215556839496658720450
k 79 y 168291389391178764892633827461038721099122240312718413098722750999543280902311723560160591
j 91 x 405128679519782118036327857853746812803438499544842440517254615347337004397796163128453499
k 80 y 2343996673345769628385945035034924111898556688243984897783198310104722693725725859068230704
j 92 x 4010360487580909976094635730099873120408451570577190233729971389454257828421122134625814540
k 81 y 32647662037449596032510596663027898845480671395103070155866053590466574431257850303395069265
j 93 x 39698476196289317642910029443144984391281077206227059896782459279195241279813425183129691901
k 82 y 454723271850948574826762408247355659724830842843198997284341551956427319343884178388462739006
j 95 x 3890045538556833346887146557570354723532742127710707027444163754745786308417317871783581352799
k 83 y 6333478143875830451542163118799951337302151128409682891824915673799515896383120647135083276819
j 96 x 38507480984093021202418459917002197264535018956615376865707542926055364929203465588139142423520
k 84 y 88213970742410677746763521254951963062505284954892361488264477881236795230019804881502703136460
j 97 x 381184764302373378677297452612451617921817447438443061629631265505807862983617338009607842882401
k 85 y 1228662112249873658003147134450527531537771838240083377943877774663515617323894147693902760633621
j 98 x 3773340162039640765570556066207513981953639455427815239430605112132023264906969914507939286400490
k 86 y 17113055600755820534297296361052433478466300450406274929726024367407981847304498262833135945734234
j 99 x 37352216856094034277028263209462688201614577106839709332676419855814424786086081807069785021122499
k 87 y 238354116298331613822159001920283541166990434467447765638220463369048230244939081531970000479645655
j 100 x 369748828398900702004712076028419368034192131612969278087333593446012224595953848156189910924824500
k 88 y 3319844572575886772975928730522917142859399782093862444005360462799267241581842643184746870769304936
Thu May 3 10:02:52 PDT 2018
==================================
int main()
{
system("date");
mpz_class x, xx, xxx;
mpz_class y,yy,yyy;
x = 10; xx = 1;
y = 14; yy = 1;
mpz_class j,k;
j = 2;
k = 2;
int goon = 1;
while ( goon && j < 100)
{
while (goon && x < y)
{
xxx = 10 * x - xx ;
xx = x;
x = xxx;
++j;
}
cout << " j " << j << " x " << x << endl;
if( x == y)
{
cout << " WOW " << endl;
goon = 0 ;
}
while( goon && y < x)
{
yyy = 14 * y - yy;
yy = y;
y = yyy;
++k ;
}
cout << " k " << k << " y " << y << endl;
if( x == y)
{
cout << " WOW " << endl;
goon = 0 ;
}
}
system("date");
return 0;
}
==================================
The problem is related to the Cannonball problem in Wikipedia. In more detail, the book Unsolved Problems in Number Theory Second Edition by Richard Guy in section D3 starting on page 223:
The case "square pyramid = square" is Lucas's problem. Is $\, x=24,\ y=70 \,$ the only nontrivial solution of the diophantine equation $$ y^2 = x(x+1)(2x+1)/6? $$ This was solved affirmatively by Watson, using elliptic functions, and by Ljunggren, using a Pell equation in a quadratic field. Mordell asked if there was an elementary proof, and affirmative answers have been given by Ma, by Xu & Cao, by Anglin and by Pinter.
The same equation in disguise is to ask if $\, (48,140) \,$ is the unique nontrivial solution to the case "square = tetrahedron", since the previous equation may be written $$ (2y)^2 = 2x(2x+1)(2x+2)/6, $$ though, as Peter Montgomery notes, this doesn't eliminate the possibility of an odd square. A more modern treatment is to put $\, 12x = X-6,\ 72y = Y \,$ and note that $\, Y^2 = X^3 - 36X \,$ is curve $576H2$ in John Cremona's tables. The point $\, (12,36) \,$ (which gives an odd square) serves as a generator. There's an infinity of rational solutions, but the only nontrivial integer solution to the original equation is given by the point $\, (294,5040). \,$
The modern treatment that Richard Guy is writing about here uses the elliptic curve with Cremona label $\texttt{"576h2"}$ or LMFDB label 576.c3 which has two $2$-torsion point generators $(0,0)$ and $(6,0)$. The point $(0,0)$ added to $(-3,9)$ is $(12,36)$ where the latter two points are infinite order generators. The other integral points are $\ (-6,0), (-2,8), (18,72), (294,5040). \ $ This is directly related to the congruent number problem. In particular, finding rational right triangles with area $6$. Read about this in, for example, Karl Rubin on Right triangles and elliptic curves. Another method is given in my work at Weierstrass Elliptic Function Polynomials.
In brief, I construct four polynomial sequences $\, (w_n,x_n,y_n,z_n) \,$ such that $\, (x_n/x_0)^2 - (y_n/y_0)^2 = (x_1^2-y_1^2) (w_n/w_1)^2, \,$ $\, (z_n/z_0)^2 - (y_n/y_0)^2 = (z_1^2-y_1^2) (w_n/w_1)^2, \,$ $\, (z_n/z_0)^2 - (z_n/z_0)^2 = (z_1^2-x_1^2) (w_n/w_1)^2 \,$ for all integer $\, n.$ They satisfy recursions such as $\, w_{n+1}w_{n-1}x_0^2 = w_n^2 x_1^2 - x_n^2 w_1^2, \,$ and $\, x_{n+1}x_{n-1}x_0^2 = x_n^2 x_1^2 - (x_1^2-y_1^2)(x_1^2-z_1^2)(w_n/w_1)^2 x_0^4. $
To get solutions of $\, j^2 = 1 + 24m^2, \, k^2 = 1 + 48m^2 \,$ we let $\, x_0 = y_0 = z_0 = w_1 = x_1 = 1 \,$ and $\, y_1 = \sqrt{2}, \, z_1 = \sqrt{3}. \,$ This is almost exactly what I used in my answer to MSE question 2556314. Let $\, t_n := w_n/\sqrt{24}. \,$ The solutions are given by $\, j = y_{2n}/x_{2n}, \, k = z_{2n}/x_{2n}, \, m = t_{2n}/x_{2n}. \,$ For $\, n=0, \,$ we have a trivial solution $\, (1)^2 = 1 + 24(0)^2, \, (1)^2 = 1 + 48(0)^2. \,$ For $\, n=1, \,$ we have a nontrivial solution $\, (-5)^2 = 1 + 24(-1)^2, \, (-7)^2 = 1 + 48(-1)^2. \,$ For $\, n=2, \,$ we have $\, (-1201/1151)^2 = 1+24(70/1151)^2, \, (-1249/1151)^2 = 1+48(70/1151)^2. \, $ For all $\, n>1 \,$ the solutions are non-integer rationals.
Now the cannonball connection is that solutions of $\, y^2 \!=\! x(x\!+\!1)(2x\!+\!1)/6 \,$ come from letting $\, x\!=\!24m^2, \, x\!+\!1\!=\!j^2, \, 2x\!+\!1\!=\!k^2, \, y \!=\! 2mjk \,$ from solutions of $\, j^2 \!=\! 1 \!+\! 24m^2, \, k^2 \!=\! 1 \!+\! 48m^2 .\,$ Thus, if only one nontrivial integer solution of this equation exists, then the cannonball problem has only one nontrivial solution and this has been proven by several authors and methods.
Not an answer yet, only a comment. I was not able to arrive at an answer but this work might give someone else more experienced than me an idea how to proceed
Step a: Let's denote the expression with the multiplicator $24$ as the diophantine equation
$$ 24x^2 +1 = t^2$$
and we look at the set of $x_h$ with natural index $h \ge 0$ such that we get a set of perfect squares of $t_h$
That set is $X=\{0,1,10,99,980,... x_h,...\}$ and we can find
the recursion-formula $x_{h+2}=10 x_{h+1} - 1 x_h $ beginning with $x_0=0,x_1=1$ or
the $\sinh()$ formula using the following constants: $s_q=\sqrt{5^2-1}$ , $q = 5 + s_q$, $\quad l_q= \ln q$ and then $$x_h=f(h)=\sinh(l_q \cdot h)/s_q $$
Step b: We can do the same with the expression with $48$ as multiplicator:
$$ 48y^2 +1 = u^2$$
and we look at the set of $y_i$ with natural index $i \ge 0$ such that we get a set of perfect squares of $u_i$
That set is $Y=\{0,1,14,195,2716,... y_i,...\}$ and we can find
the recursion-formula $y_{i+2}=14 \cdot y_{i+1} - 1 y_i $ beginning with $y_0=0,y_1=1$ or
the $\sinh()$ formula using the following constants: $s_r=\sqrt{7^2-1}$ , $r = 7 + s_r$, $\quad l_r= \ln r$ and then $$y_i=g(i)=\sinh(l_r \cdot i)/s_r $$
Step c: Now I see two options, but I can't finish any of them.
Opt c1: Using the $\sinh()$ formula one can formulate
$$ get\_h(i)\quad = f°^{-1}(g(i)) \qquad= \sinh°^{-1}( \sinh(l_r \cdot i)/s_r \cdot s_q)/l_q $$
This gives the fractional index into a generalized set $X$ (when generalized to the reals) for any $y_i$ at integer index $i$ in $Y$.
Computing the generalized $h$ from the integer $i$ such that $x_h = y_i$ by $h=get\_h(i)$ gives some heuristical data:
(update: added more information in following table)
i h(generalized) i'=(h+d_24)/c_24 err1=i'-i h'=i*c_24-d_24 err2 = h - h'
-----------------------------------------------------------------------------------------------------
0 0.0 0.13158112005 0.131581120053 -0.151181644749 0.151181644749
1 1.00000000000 1.00193228121 0.00193228120785 0.997779883231 0.00222011676906
2 2.14675300150 2.00001008762 0.0000100876239708 2.14674141121 0.0000115902918512
3 3.29570299894 3.00000005200 0.0000000520028797854 3.29570293919 0.0000000597493082176
4 4.44466446748 4.00000000027 0.000000000268063295398 4.44466446717 0.000000000307994413475
5 5.59362599515 5.00000000000 1.38180628232E-12 5.59362599515 1.58764225751E-12
6 6.74258752313 6.00000000000 7.12290205824E-15 6.74258752313 8.18394043250E-15
7 7.89154905111 7.00000000000 3.67169655979E-17 7.89154905111 4.21863808961E-17
8 9.04051057909 8.00000000000 1.89267738302E-19 9.04051057909 2.17461349797E-19
9 10.1894721071 9.00000000000 9.75632822017E-22 10.1894721071 1.12096457793E-21
10 11.3384336351 10.0000000000 5.02916879513E-24 11.3384336351 5.77832146333E-24
11 12.4873951630 11.0000000000 2.59242393236E-26 12.4873951630 2.97859536250E-26
12 13.6363566910 12.0000000000 1.33633650388E-28 13.6363566910 1.53539923139E-28
13 14.7853182190 13.0000000000 6.88851552903E-31 14.7853182190 7.91463932775E-31
14 15.9342797470 14.0000000000 3.55087555088E-33 15.9342797470 4.07981939861E-33
15 17.0832412750 15.0000000000 1.83039685760E-35 17.0832412750 2.10305557032E-35
16 18.2322028029 16.0000000000 9.43528605354E-38 18.2322028029 1.08407806810E-37
17 19.3811643309 17.0000000000 4.86367874500E-40 19.3811643309 5.58817976246E-40
18 20.5301258589 18.0000000000 2.50711751613E-42 20.5301258589 2.88058157216E-42
19 21.6790873869 19.0000000000 1.29236295595E-44 21.6790873869 1.48487531658E-44
20 22.8280489149 20.0000000000 6.66184173327E-47 22.8280489149 7.65419985702E-47
21 23.9770104428 21.0000000000 3.43403028343E-49 23.9770104428 3.94556868158E-49
22 25.1259719708 22.0000000000 1.77016573789E-51 25.1259719708 2.03385233098E-51
23 26.2749334988 23.0000000000 9.12480811460E-54 26.2749334988 1.04840534739E-53
24 27.4238950268 24.0000000000 4.70363431774E-56 27.4238950268 5.40429487278E-56
25 28.5728565548 25.0000000000 2.42461819659E-58 28.5728565548 2.78579302792E-58
26 29.7218180827 26.0000000000 1.24983640354E-60 29.7218180827 1.43601394393E-60
27 30.8707796107 27.0000000000 6.44262687544E-63 30.8707796107 7.40233041901E-63
28 32.0197411387 28.0000000000 3.32102993149E-65 32.0197411387 3.81573562456E-65
29 33.1687026667 29.0000000000 1.71191658606E-67 33.1687026667 1.96692629650E-67
30 34.3176641947 30.0000000000 8.82454677642E-70 34.3176641947 1.01390647480E-69
Here, the constant $c_{24}$ is $c_{24}=f°^{-1}(\sqrt 2)$ and $d_{24}=f°^{-1}(\sqrt 3 / 24)$.
We see in column 3 (i'), that a simple linear transformation of the generalized $h$ by the (irrational) constants $c_{24}$ and $d_{24}$ is enough to reproduce integer $i$ with a quickly vanishing error.
In reverse, the values $h'$ taken by the linear function $h'(i)=i \cdot c_{24}+d_{24}$ approximates $h(i)$ very quickly with vanishing error. Since $c_{24},d_{24}$ are likely transcendent values it should be impossible to equal an integer at any index $i$.
With this I tried to show, that the function $get\_h(i)$ is non-integer for all integer $i>0$ - but of course, such a conclusion needs now some more formal arguing using arguments of from theory of diophantine approximation.
Opt c2: We want, that at some $h>0$ and $i>0$ we find (at least) one equality $x_h = y_i$. Here we observe, that the primefactorization in $x_h$ and $y_i$ is cyclic with the indexes and each possible primefactors in $x_h=y_i$ has its specific cycle-length. The cycle-lengthes of the primefactors in $x_h$ correspond now with primefactors in $h$ and that of $y_i$ with primefactors in $i$.
Here is now the task to show that the cycle-lengthes for $x_h$ and $y_i$ are different enough so that we do not find any combination of $h$ and $i$ such that $x_h=y_i$ by their canonical primefactorization.
This allows to exclude very large $x_h$ and $y_i$ with little effort, but again I do not have a conclusive method to show that this has, so to say, an "infinite ascent".
Note, this comment is reflecting a related question of mine and by this a very similar question having only different parameters and which I've tried to answer the same way as written here.